Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) . Divide knot presentation of Berge's knots . of lens space surgery . .. Yuichi YAMADA (Univ. of Electro-Comm. Tokyo) Singularities, knots, and mapping class groups in memory of Bernard Perron 2010 Sept. Univ. of Bourgogne, Dijon, France 1 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) . ..1 Introduction. Dehn surgery . ..2 Divide knots . ..3 Lens space surgery . ..4 Results. (Divide presentation of Berge knots) 2 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) x1. Dehn surgery Dehn surgery = Cut and paste of a soliod torus. ( ) 3 (K; p) := S n open nbdN(K) [@ Solid torus: Coefficient (in Z) \framing" = a parallel curve (⊂ @N(K)) of K, or the linking number. Solid torus is reglued such as \the meridian comes to the parallel" 2 = 3 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) . Theorem ([Lickorish '62]) . .. Any closed connected oriented 3-manifold M is obtained by a framed link (L; p) in S3, ie, M = (L; p), [ [···[ . (L; p) = (K1; p1) (K2; p2) (Kn; pn): .. Lens space L(p; q) p 1 = a1 − (ai > 1) q 1 a2 − .. 1 a3 − . − an n − p −a1 −a2 −a3 −an q r n 1 = ··· − r For n 2 Z; r 2 Q 4 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) . Theorem (Kirby-Rolfsen moves (Fenn-Rourke's ver.)) . .. ∼ 0 0 The 3-manifolds are homeo. (L; p) = (L ; p ) , framed links (L; p); (L0; p0) are moved to each other by isotopy .and the following .. Note: This (with a suitable sign) is blow-down/up, related to resolution of the singularity. The green curve (axis of the twist) is called the exceptional curve. 5 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) Starting example of lens space surgery [Fintushel{Stern '80] It is not easy to find/prove \unexpected" lens space surgery. What is the best method to prove it? 6 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) My answer ([Y]) : blue [ green = S3, and red becomes the knot P(−2; 3; 7). The knot is constructed by seq. of full-twists = blow-downs. 7 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) Knots Singularities are constructed , are reslolved by some full-twists by some blow-ups In 2000, I heard A'Campo's divide theory: a generic plane Curve P ) a Link L(P) in S3 A'Campo's divide knots ['75] |{ Plane curves are easier to draw than knots. 8 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) Typical example of my results. 9 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) . Theorem (Main Theorem) . .. Every Berge's knot of lens space surgery is a divide knot. .. Berge's list ('90), is believed to be the complete list of lens space surgery. For the proof, it took many years. Because I am lazy, but As a phenomenon, lens space surgery is not simple. The set (list) consists of three subfamilies of inifinite knots. Each subfamily has each \personality". At first, I hoped to synthesize all of them deductively. But after all, Each subfamily needs each method in the detail. 10 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) By another approach (Heegarrd Floer homology, C-links by Rudolph ··· ), it is proved: . Lemma (Hedden) . .. Any knots of lens space surgery is intersection of an algebraic .surface in C2 and a 4-ball. .. My study is more concrete, to know · \How"does each knot yield a lens space? · the construction of each knot of lens space surgery, · the set of lens space surgeries. ··· 11 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) x2. A'Campo's divide knots Original construction. a generic plane Curve P ) a Link L(P) in S3 A'Campo's divide knots Let P be a generic (no self-tangency) curve in the unit disk D, 3 2 2 S = f(u; v) 2 TDju 2 D; v 2 TuD; juj + jvj = 1g 2 2 3 L(P) := f(u; v) 2 TDju 2 P; v 2 TuP; juj + jvj = 1g ⊂ S : u v P D · Here is the strongly-involution ι :(u; v) 7! (u; −v). 12 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) Ex. Torus links ['02 Goda-Hirasawa-Y, (Gusein-Zade, etc.)] P is the p × q rectangle (billiard) curve ) L(P) is T (p; q) (PL curve with slope 1 in the rectangle.) ex. (p; q) = (7; 4) 7 4 (p − 1)(q − 1) It has double points, in general. 2 13 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) Basics on Divide knots [N.A'Campo, L.Rudolph,..] (0) L(P) is a knot (]L(P) = 1) , P is an immersed arc. (1) The genus of knot L(P) = ] double points of P. (2) lk(L(P1); L(P2)) = ](P1 \ P2), if they are knots. (3) Every divide knot L(P) is fibered. (4) Any divide knot is a closure of strongly quasi-positive braid. ie, product of some σij . (5) P1 ∼ P2 by ∆-move ) L(P1) = L(P2). L is not injective. σij ∼ i j ɺ+kmtcmlnj_lcasptcq 14 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) These curves present the same knot P(−2; 3; 7) (Thanks to Hirasawa) Only 19 and 18 are the coefficients of lens space surgery. 15 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) \Ordered Morse divide (OMD)" [Couture] A divide is called OMD if (w.r.t at least one direction) max/min points are in the same level, up to isotopy. ( This is a Non-OMD. (w.r.t horizontal nor vertical direction) 16 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) ['01 Couture-Perron] proved a visualized version an OMD P ) a knot L(P) in S3 Braid presentation of L(P) In this talk, we demonstrate them by P(−2; 3; 7) 17 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) ['01 Couture-Perron] proved a visualized version an OMD P ) a knot L(P) in S3 Braid presentation of L(P) The braid presentation of L(P). mp → , , mp → → We get the fiber surface of L(P). genus = 5. 18 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) 3 → 2 1 σ3σ2σ1 σ3σ2σ1 σ3σ2σ1 σ2σ1 σ2σ1 3 2 The result is W4 W3 , where Wn := σn−1σn−2 ··· σ2σ1 \1=n twist". 3 Exercise. Show that W5 W3 induces the same knot. 19 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) We focus on curves cut from the lattice Assume (on the regions) · Vertices are in Z2 ⊂ R2, · Edges of the region are vertical or horizontal, · Every concave point is at odd point, and · The curve P is an immersed arc (, L(P) is a knot). where we call a point (m; n) 2 Z2 odd if m + n ≡ 1 mod 2 20 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) Examples. In fact, these examples presents knots of lens space surgery. · L-shaped curve · Generalized L-shaped curve 0 / . / 0 · T-shaped curve · Generalized T-shaped curve T (5, 6) T (6, 7) 35 48 21 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) . Lemma (Y) . .. \Adding a square" corresponds to a right-handed full-twist = blow-down = coord. transform: (x; y) = (X ; Y =X ). 2 2 2 .(ex. y = x + ϵ becomes Y = X (X + ϵ)) .. y Y x Ŵ X ∼ Ŵ ∆ @jmubmul Ŵ 22 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) Cable knot: C(T (a; b); p; pab + r) of a torus knot is a divide knot. ex. C(T (2; 3); 2; 13) pa pb r p { { x = ta x = tap It is algebraic: From T (a; b) = to , or y = tb y = tbp + tbp+r ( ) b r y = x a 1 + x ap Puiseux pair is f(b; a); (bp + r; p)g. 23 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) More iterated cables () generalized L-shaped) C( C(T (a; b); p; pab + r); P; Pp(pab + r) +( R)) is represented,( )) b r R (ex. C( C(T (2; 3); 2; 13); 3; 80)), y = x a 1 + x ap 1 + x apP . pa pb r p ×P Ppa Ppb Pr R Pp P 24 / 41 Introduction. Dehn surgery Divide knots Lens space surgery Results. (Divide presentation of Berge knots) Appplication : Twisted torus knot T (p; q; r; s) is constructed from T (p; q) (in the standard position) by s full-twists of the r strings in the p strings. → T (5, 2) T (5, 2; 4, 1) Fact. T (p; q; r; s) can be cable knots, for specially controlled p; q; r; s ([Morimoto-Y], SangYop Lee) 25 / 41 Introduction.